= Solution
Let $D=d\mathbb Q/d\mathbb P$ on $\mathcal F$. The <Radon-Nikodym derivative> is nonnegative and $\mathbb E_{\mathbb P}D=1$. For $A\in\mathcal F_t$,
$$
\mathbb Q(A)=\mathbb E_{\mathbb P}(D\mathbf1_A)=\mathbb E_{\mathbb P}\bigl(\mathbb E_{\mathbb P}(D\mid\mathcal F_t)\mathbf1_A\bigr).
$$
Uniqueness of the restricted <Radon-Nikodym derivative> therefore identifies
$$
\boxed{Z_t=\mathbb E_{\mathbb P}(D\mid\mathcal F_t)\geq0.}
$$
For $s\leq t$, the tower property of <conditional expectation> gives $\mathbb E_{\mathbb P}(Z_t\mid\mathcal F_s)=Z_s$, and $\mathbb E_{\mathbb P}Z_t=1$. Thus $Z$ is a nonnegative true <martingale>, indeed a <uniformly integrable martingale>, since it consists of conditional expectations of one integrable random variable. This is the <Radon-Nikodym density martingale>. Under the <usual conditions for a filtration>, it has the standard càdlàg version; continuity is the additional assumption in part (c), not a consequence of absolute continuity alone.
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